Mathematical Proof of String Theory

I have only taken a few college physics courses, and I have watched some videos about string theory. I would like to get excited about M-theory, but I have always wondered, why there wouldn't be an infinite number of ways to unite the forces, in which case M-theory is just one possibility amongst an infinite number of possibilities.

String theory or M-theory is one mathematical "mechanism" which unites all the forces, Garret Lisi's theory is another "mechanism" which unites the forces into a coherent mathematical structure.

This means that there are at least 2 mathematical mechanisms of uniting the forces. But, why stop at an arbitrary number like 2? What evidence do we have that there are not literally an infinite number of ways to unite the forces? All you need is to propose some mathematical "mechanism" which allows the forces to be compatible, and with math being infinite like it is, I see no reason why there would not be an infinite number of ways which can unite the forces.

If this is the case, it would seem very premature to get excited about any individual theory, when there are infinite possibilities.

If it were possible to mathematically prove, or establish boundaries on the type or number of mechanisms, then it might be possible to show that M-theory is the only possible way to mathematically unite the forces, which would essentially be a mathematical proof that M-theory is true, since we obviously assume our universe must obey the laws of math.

One way to do this would be to mathematically prove that any method of mathematically uniting the forces would necessarily make use of strings or banes. This would allow us to know for a fact that our universe is made up of strings, even though they are too small to see.

Can you give any information about the points I raised above? And just to simplify, my main question is:

What mathematical research has been done on the number of "mechanisms" which can unite the forces?

It is quite simple to say that there are a multitude of ways to mathematically construct an idea that would unify the forces of nature. In fact there are continuous efforts for mathematicians and Physicists to find a pure and unflawed theory for instance Albert Einstein, Theodore Kaluza, and Ancient Greek Atomists all developed theories to find the unifying theory. The only problem is there are very few that contain the mathematical consistency that theories like M-Theory do and have the depth and ambiguity that allows us to see more thoroughly into the structure of the Universe. M-Theory is rather pre-mature for its time it will be long before we understand it thoroughly and when we do I'm quite certain that it will convey something beyond comprehension. So yes, there are 'infinite' theories that do exist but only few like M-Theory contain the elegance, complexity and consistency that make it a probable Theory of Everything

I saw this video, where Edward Witten says that "there were a lot less theories that were possible" when talking about string theory http://www.youtube.com/watch?v=iLZKqGbNfck#t=3m30s", so is he talking about ANY theory, or only theories which use strings?

It seems to me that scientists must have narrowed it down to the point where string theories are the only possibilities, otherwise why would they be so excited about string theory, if there is infinite other possibilities which will unite the forces in a coherent mathematical structure?

And then he explains that the 5 possible string theories are actually limiting boundary conditions on the M-Theory http://www.youtube.com/watch?v=iLZKqGbNfck#t=5m30s", again, is he saying that ANY possible theory will have to fall somewhere in that shape, or is he saying that theories using strings as the basis will have to fall in that shape?

He's most likely referring to the String Theory Landscape which is the great amount of false vacua that arise due to the configurations and "choices of Calabi-Yau Manifolds and different values of generalized magnetic fluxes over different homology cycles" - Wikipedia: http://en.wikipedia.org/wiki/String_theory_landscape
It is believed amongst String Theorists that the 10500 possible configurations of the false vacua contain atleast one in which is structured by the laws of General Relativity and Quantum Mechanics and contains a small and positive cosmological constant ([tex]\Lambda[/tex]) and is suitable for generating human life. Some String Theorists use the Anthropic Principle too narrow the plausible false vacua.

Well there's ofcourse the paining task of being empirical and predictable in physics standards, which seems to be more important, as its a model of physical reality, than being mathematically well structured.

I've not read of any specific results regarding the number of mechanism that can unite the forces. If no one cites any references, maybe you could try the mathematics section of the forums....

I also wonder if that is what you really mean because I would instead like to know the mechanism(s) to unify not just forces, but space, time, energy, as well. I don't personally feel that forces arise via one set of independent mechanisms and other physical components of our universe from others. Seems like we should be able to find a common origin.

Roger Penrose is one guy who has his feet in both mathematics and physics. His ROAD TO REALITY might offer you some clues. I've read maybe half of it but past page 100 or so the math is beyond my paygrade and I only understand bits and pieces. If you use the Index at the back of the book and look under MATHEMATICS you'll find some relevant reading. But not a specific answer to your question I don't think.

For example Penrose discusses how compactness of space leads to the discreteness of quantum theory....so depending on the approach you take, there are a number of physical factors that may restrict what at first appears to be an infinte number of choices.

Another limit/constraint is referred to by Kip Thorne in BLACKHOLES AND TIME WARPS: On page 453 he mentions physicsts become very suspicious when their equations indicate an infinity...he says it is not currently believed there are any infinities in the universe....so that suggests that neither general relativity nor quantum mechanics, which diverge at the big bang and black hole singularities, is quite the righjt answer.

Some constaints on the number of theories: they have to be self consistent, match physical observational experiments,account for the spin of force carriers (photons, gravitons,etc) and on and on...I'm sure someone who is up to date on math could make a long list of such constraints....

Pure speculation:
The other side of the coin just might be that if there really are an infinite number of universes, as in parallel universes for example, maybe there is an infinite number of originating mechanisms....wouldn't that be cool. That's not a requirement for having an infinite number of universes, but would provide some fascinating insights. There is a thread around here that discusses the many different types of infinites that exist in mathematics, so I would not automatically disregard the possibility that maybe one of those "infinities" is a generation mechanism for universes...

It is a nice idea that a mathematician could sit down and work out the set of all possible theories consistent with known physics.

I fear this rather exagerates the ability of theoretical physicists (for all their genius).

String theory developed by a number of relatively small steps from the standard methods of quantum field theory. This is very striking when you read an elementary text on string theory. If you understand quantum field theory you'll have no problem learning string theory.

So string theory and M-theory have come about through a process of inspired stumbling. If nature is kind to us then this is the answer, but there is no a priori reason why it should be.

I am an evolutionary psychologist (Darwinist), yet have always loved the elegance of physics (though most of it is beyond my comprehension). I would like to "bounce off" all you geniuses out there my simple, and somewhat novel explication of what, I believe, Superstring theory represents ----to me, at least. It also causes me to be convinced that the basic idea of the theory pretty much has to be right!

I'm also going to state this as I would to another non-physicist sitting in a chair next to me (by the fire, as it were). I want to know if I should continue to explain it this way or not.

I would suggest that Superstring Theory represents the resolution of a confusion about the relationship of mathematics to physics. Physics requires some physical extension in "reality"; whereas math is pure theory. In math, one commonly utilizes the idea of dimensionless points. Physics, which is so used to importing ideas from math, seems to have forgotten the philosophical difference between the 2 disciplines. It began utilizing the idea of "point-particles" ---which were conceived of as "dimensionless"; i.e. without physical extension. To my mind, with Superstring theory physicists (through extremely difficult mathematics) finally "came to their senses".

Physics is a discipline, unlike mathematics, where some physical extension is absolutely necessary. Thus, it was never possible that there could have been such thing as "dimensionless" "point" particles. Though subatomic "particles" are conceived to be the smallest possible "things", they still need to have some physical extension in order to "exist" in "reality". Superstring Theory, philosophically, recognizes that the smallest possible physical extension must be a one-dimensional entity (once you accept that "dimensionless" cannot be a physics construct, there must be at least one physical dimension in play). Yet, of course, the most basic physical entity must also be the "smallest possible" one-dimensional entity.

Now, most people would believe that there is no such thing as the smallest possible thing. One would logically assume that "you can always cut it in half" and get something smaller. This logical assumption is very similar to the belief that, however fast something is moving, it can always go a little faster. Einstein proved, at the time shockingly, that the latter assumption was wrong. You cannot always go a little faster. The universe has a "speed limit", and nothing can ever accelerate to the speed of light. Yet few ordinary people realize that, just like the false assumption about speed, so our assumption that there is no limit to smallness is also wrong.

All physicists realize that the is a limit to smallness ---it is called Planck Length. So, at last, we can reasonably talk about "least possible physical extension" as a meaningful physics construct replacing the construct of "dimensionless" mistakenly imported from mathematics. The simplest physics entity that could ever "exist" would be a one-dimensional entity of Planck Length. Is it, then, surprising that this turns out to be the very definition of a Superstring? (Then I go on to make the standard metaphors about subatomic entities being like "notes" played on the tiniest string vibrating in 6 dimensions within 11 dimensional hyperspace)

Anyways, that's the simple way I think of Superstring Theory such that it seems to me so obviously right. Can I go on explaining it to my friends in this manner, or is this perspective wrong?

Physics is a discipline, unlike mathematics, where some physical extension is absolutely necessary. Thus, it was never possible that there could have been such thing as "dimensionless" "point" particles. Though subatomic "particles" are conceived to be the smallest possible "things", they still need to have some physical extension in order to "exist" in "reality".

Why do you say this ?
I suspect that that the point particle notion will be replaced eventually by something - maybe strings, maybe not. I don't know. However, I wouldn't go as far as saying that particles cannot be modelled as zero dimensional objects simply because I don't like that idea.

Why do you say this ?
I suspect that that the point particle notion will be replaced eventually by something - maybe strings, maybe not. I don't know. However, I wouldn't go as far as saying that particles cannot be modelled as zero dimensional objects simply because I don't like that idea.

Do you have a specific reason for rejecting point particles ?

I would contend that the notion of indivisible "point-like particles" has long been non-existent. First, In ordinary quantum mechanics, before quantization we have point-like particles, but after quantization we only have the "cloudy" wavefunction.

Second, In quantum field theory, even before quantization we only have fields. In a field you can make wavepackets of arbitrarily small size but not zero size - therefore there's no "point-like" particles. In other words, you can approach a point using Fourier modes of higher and higher energy (never mind if there's any cut-off) but never reach a perfect dimensionless point.

So my take is that we have been dealing with extended objects for a long time. There's no need for anyone to feel that our theories of physics assume the existence of unintuitive, indivisible, mathematically perfectly dimensionless point-objects. Therefore I don't agree with the opinion that string theory is conceptually more satisfying because it removes "point-like" particles.

Even without going into technical details (we renormalize the electron-photon vertex starting from the assumption of a point-like interaction for instance, compare with say the proton-photon vertex), do you agree that we can experimentally address the question of the electron size, and that we have today no evidence of such an extension down to ~ [itex]10^{-20}[/itex] m ?

The argument I made may have other flaws, but I never said I did not "like" the idea of dimensionless points. I gave my reasons for why I think it makes perfect sense to use dimensionless points as mathematical constructs that can represent anything from scampering beetles to jet planes. Yet when one is choosing a model for what is the deepest essence of physicality, a dimensionless point is not feasible. It is, in fact, its very opposite ----a dimensionless point could reasonably described as "the essence of non-physicality"!

And I'm not saying a lot of physicists didn't successfully utilize the dimensionless point model when they couldn't more deeply describe what was the most basic entity of physical reality. Certainly, they needed a "place-holder" of some sort for purposes of describing subatomic interactions. It's only when they began thinking that a dimensionless point really was an appropriate description of that essence that they got into trouble ---namely, the singularity.

In a sense, the phenomenon of the singularity is a kind of proof of my argument. It effectively demonstrates the logical futility of utilizing the mistaken idea that a dimensionless point as a good descriptor for the essence of physical (not mathematical) reality.

So again, my position would be that for something to exist in physical reality, there must be some physical extension. And superstrings represent the best possible model for the very reason that they are the least possible physical extension.

My prior response was only with regards to Sheaf. With regards to Petergreat's conclusion ("...we have been dealing with extended objects for a long time"), I can only say that isn't it both likely and logical that the ultimate descriptor of essence (given that physics has always, as you say, been dealing with extended objects) would be the least possible extension ---i.e. a one dimensional entity of Planck Length. I know I'm taking advantage of hindsight, but once the superstring was mathematically conceptualized, shouldn't everyone have said, in effect, "But of course! What else could it have been?" Finally, with regard to Humanino, I must admit that such arguments, while perhaps the ultimate foil to my logic, are simply beyond my understanding of physics. I apologize for not being capable of addressing his point.

In any event, my whole point was to stimulate discussion and watch and learn from your responses. Thank you one and all.